Sequences

.        Introduction

.        Terminology

.        Some Common Sequences

.        Linear Sequences

.        Position-Position Rule

.        Position-Term Rule

.        Times Tables As Sequences

.        A Matchstick Pattern Application

.        Quadratic Sequences

.        An Example In Using A Quadratic Sequence

.        Geometric Sequences

.        Fibonacci Sequence

Introduction

Simply put, a sequence is a (usually) a list of numbers. These are separated by a comma and may or may not continue forever. Usually we hope there is a rule which links the numbers to each other. However it is possible to have a random sequence in which there is, of course, no rule by definition.

Terminology

There is some terminology associated with sequences which we need to know.

Here is a typical(?) sequence: 3, 7, 11, 15, ...

Each number is called a term. So 3 is a term, 7 is a term and so on. Actually 3 is the first term (1st term), 7 is the second term (2nd term) and so on.

Notice that the terms are separated by commas. At the end we have the typographical symbol called an ellipsis, the three dots, which is used to indicate that the terms continue on and on forever.

Each term occupies a position in the sequence, so we can see that the three occupies the first position or position one, the seven occupies the second position or position two etc.

One thing we can investigate to our advantage, especially for the simpler sequences we encounter, is the difference between the terms. Actually, we specifically look at the term in the higher position and subtract the previous term. So in our sequence above we get 7 - 3 = 4. So the first difference between the first two terms is +4. In fact we will find this is the difference between all the successive terms for this sequence.

Note: We can also add the terms together. In this case we have what is called a series.

Some Common And Special Sequences

A constant sequence:

3, 3, 3, 3, ...

The same number all the time

The even numbers:

2, 4, 6, 8, 10, ...

Go up in twos

The odd numbers

1, 3, 5, 7, 9, 11, ...

Go up in twos, but starts at 1

The prime numbers

2, 3, 5, 7, 11, 13, 17,...

What rule links these?

The square numbers

1, 4, 9, 16, 25, 36, ...

Each term is its position squared

A linear sequence

3, 7, 11, 15, ...

The difference between the terms is constant (=4 in this case)

A geometric sequence

1, 2, 4, 8, 16, 32, ...

Each term is a constant multiple of the previous term - this is the doubling sequence.

A quadratic sequence

4, 5, 8, 13, 20, 29, ...

Discussed later, but look at the differences.

The triangular numbers

1, 3, 6, 10, 15, 21, ...

Actually a quadratic type of sequence

The Fibonacci numbers

1, 1, 2, 3, 5, 8, 13, 21, ...

Each term is the sum of the previous two terms.

There are many other types of sequence: exponential, logarithmic, power, trigonometric, Taylor, etc.

Linear Sequences

The feature that defines a sequence as being a linear sequence is that the difference between successive terms is always the same value, a constant. For instance:

1, 2, 3, 4, 5, ... the difference between successive terms is +1.

4, 7, 10, 13, ... the difference between successive terms is + 3

10, 9.5, 9.0, 8.5, 8.0, ... the difference between successive terms is -0.5

One of our main aims in working with sequences is to try and determine the value of any term at any position we state. Another aim might be to decide if a particular value is actually in a given sequence.

Term To Term Rule

There are two ways we can work out our terms. The first one we’ll look at is to work out the term from the previous term or terms.

For instance suppose we are given the sequence

4, 7, 10, 13, ...

We can see that the difference is +3. So our rule to get the next term is to add 3 to the previous term.

Here is another one:

10, 6, 2, -2, ...

The next term is the previous term subtract 4.

We need now to use a more mathematical notation.

We can call the term at position n, Tn This is just read: “tee of en or tee at en”.

Using this notation we can call the term at position 1 T1, the term at position 2 T2 and so on.

The term before Tn is therefore Tn-1, and the one after Tn is Tn+1

So for the sequence 7, 12, 17, 22, ... we could say that

Tn = Tn-1 + 5

With our term to term rule, we of course have to specify at least one term in the sequence, usually the first one T1.

One of the problems with the term to term rule for linear sequences is that if I give you the first term and the rule and then ask you to find the 900th term, you are going to have to do quite a bit of work (unless you do a little thinking!).

Position To Term Rule

Another approach is to work out a position - term rule instead of a term - term rule. With this all I need to do is to provide a position and the rule will work out the value of the term for me.

To do this we need to do a little preparatory work first. So let's investigate the sequences in our ordinary times tables.

Times Tables As Sequences

The 2× table: 2, 4, 6, 8, 10, ...

Note that the difference between successive terms is 2. We, of course, expect this since the two times table is the sequence of numbers that goes up in 2's.

We can say that the term is simply the position times by 2. So that the term at the first position is 1 x 2 = 2, the term at the second position is 2 x 2 = 4, the term at the third position is 3 x 2 and so on. With this rule we can easily work out the term at any position. For example, the term at position 27 is 27 x 2 = 54. Easy.

Now let's write this using a more mathematical notation. Let Tp be the term at position p and p be the position.

For our 2× table T = p x 2, but we write this in algebra by putting the number first to get T = 2 x p or Tp = 2p

Let's look at the 7 time table. 7, 14, 21, 28, 35, ...

The difference between the terms is 7 and the formula is the position × 7 or

Tp = 7p.

This applies to any times table. The 3× table is given by the rule or formula Tp = 3p, the four times table by the formula Tp = 4p, and so on.

Let's consider again the 3× table:

3, 6, 9, 12, 15, ...

The rule is Tp = 3p.

What if we add 1 to each term? Well we would get the sequence:

4, 7, 10, 13, 16, ...

and surely the formula is Tp = 3p + 1

Here, we can see the difference between terms is still 3.

We are now in a position where we can work out the formula for any linear sequence.

Consider: 5, 7, 9, 11, 13, ...

The difference is 2, so we suggest that this sequence must be based upon the 2× table, i.e.  Tp = 2p

2, 4, 6, 8, 10, ...

We now need to ask what we do to each term in this sequence to get our desired sequence of: 5, 7, 9, 11, 13, ...

If we look at each term in turn we can see that the sequence terms are 3 more than the 2× table terms. So I have added 2 to the 2× table. Using our formula we can say that our sequence is Tp = 2p + 3

Here is another example: 6, 10, 14, 18, 22, ...

The difference is 4 so we must have Tp = 4p, the 4× table 4, 8, 12, 16, ..

And we are adding 2 to each term to get our sequence, so we must have the formula Tp = 4p + 2

If I have this formula, how do I use it? Well suppose I want to know what the value of the term at position 37 is? I can just substitute the p by 42 in the formula so T = 4x37 + 2. (Remember the multiplication is done first) This equals 150. This is the term at position 37.

A Matchstick Pattern Application

Quite often (certainly at the lower levels) we have to work with matchstick or other patterns. Here is a complete example of how such a question might work.

Here we have the first three patterns in a sequence of patterns. Each pattern is made up of matchsticks, and we can see that the pattern grows in the same way for each new pattern. If we study the patterns carefully we can see that the first pattern has three matchsticks, makes one triangle and is pattern one in the sequence. The second pattern is obtained by adding two matchstick to the first patterns so it has five matchsticks and comprises two triangles. The third pattern has another two matchsticks and has seven matchsticks and makes three triangles.

We could easily define a term to term rule for this sequence. In words we could say that we add two matchsticks to the previous pattern. In symbolic language we could say that Mn+1 = Mn + 2, where Mn is the number of matchsticks at position n.

However, it is arguably better to use the position - term idea to get a formula for the number of matchsticks. Let's call the number of matchsticks M and let's call the position by the letter P.

The actual sequence is 3, 5, 7, 9, ...

We know that the pattern goes up in twos, so is based upon the 2× table, so

M = 2P, gives 2, 4, 6, 8, ...

and each term is 1 more than the two times table:   M = 2P + 1

With this formula we can now work out how many matchsticks in any pattern. For instance how many matchsticks are there in pattern 23?

M = 2 x 23 + 1 = 47

We can also work backwards. What pattern could we make with 70 matchsticks. To do this we reverse the formula.

70 = 2P + 1,

So take off 1, leaves 69. Divide by 2, to get 34.5, so we can make pattern 34 (note we don't have enough matchsticks to make the next pattern - don't round up your answer.)

It is also useful to try and see how the formula relates to the original situation.

The 2P obviously refers to the fact that each pattern has two extra matchsticks. So pattern 1 has 1 lot of 2, patterns 2 has 2 lots of 2, pattern 3 has 3 lots of 2. Each pattern also has the one matchstick at the beginning, which is what the +1 relates to.

Quadratic Sequences

Not all sequences are linear. For instance consider this sequence:

1, 4, 9, 16, 25, 36, ...

This is the sequence of square numbers. i.e. 1×1, 2×2, 3×3 etc.

If we work out the difference between the terms we get:

To try and describe this rule in words is a little tricky. We can say that the next term is obtained by adding to the previous term the previous difference add 2. This is a clumsy sentence. Or we could say that the difference goes up by two each time and we add this to the previous term.

To try and write this in symbols is also tricky. Tn+1 = Tn + ? 

However, what do we put for the question mark, since it changes for each position.

For these types of sequences we find it easier to work with position - term definitions.

In position 1 we have T1 = 1 x 1 = 1

In position 2 we have T2 = 2 x 2 = 4

In position 3 we have T3 = 3 x 3 = 9

So in position n we have Tn = n x n = n2, so our formula is Tn = n2

If we look at our sequence we might have spotted that if we take the difference between the differences, what is called the second difference, we get 2 which is a constant.

So here we have a 2nd difference of 2 which comes from having a sequence of n2

Surely if I double the terms of the original sequence to get 2, 8, 18, 32, 50, ... I will be doubling the 2nd difference value to get 4.

The formula for this sequence is 2n2 and its second difference is 4.

In general then the coefficient of n2 is half of the second difference. (Whereas for the linear sequence the coefficient of n was the first difference.)

If the second difference = 1 then the sequence is based upon   

If we find that the second difference is a constant, then we know that the sequence is based upon n2 terms, which we call a quadratic sequence.

We can also see that in general a quadratic sequence has the form

Tn = an2 + bn + c

This is made up of two parts, the first part is Tn = an2 and the other part is the linear sequence Tn = bn + c.

So if we know that we have a quadratic sequence, because our second difference is a constant, then we can take away the an2 part from our sequence to leave a linear sequence. We can then find out what this is and by putting the two parts together we can work out our formula.

Here is an example:

5, 11, 19, 29, 41, ...

We can work out our differences

We have a 2nd difference of 2 so our sequence is based upon n2.

If we take away the terms of this we shall end up with a linear sequence

We can now work out the formula for this linear sequence. it goes up in 3's so is based on the 3x table and each term is 1 more than the 3× table. The formula is therefore Tn = 3n + 1

If we now put our two parts together we get:

Tn = n2 + 3n + 1

An Example In Using A Quadratic Sequence

A pond builder! builds square ponds and surrounds them with square tiles. The builder wants to know how many tiles are required for a particular size pond. (How contrived is this!)

Here are some sketches she made:

We can see that for a 1 by 1 pond we need 8 tiles, for a 2 by 2 pond we need 12 tiles, for a 3 by 3 pond we need 18 tiles and so on.

We have a sequence: 8, 12, 18, 26, 36, ...

The first difference is 4, then 6 then 8 and so on, the second difference is always 2.

So we know we have a quadratic and it contains n2 tiles. If we take off the n2 tiles we end up with 7, 8, 9, 10 etc. This is a linear sequence: n+6.

So our original sequence is Tn = n2 + n + 6

If the builder wants to build a 5 by 5 pond she will need 36 tiles.

Geometric Sequences

A geometric sequence is one where each term is increased by multiplying by a fixed amount (called the common ratio).

A simple example is the doubling sequence, 1, 2, 4, 8, 16, 32, 64 etc. Here the common ratio is ×2.

Note how quickly the sequence increases.

If the common ratio is ½ then the sequence will keep decreasing, e.g.

160, 80, 40, 20, 10, 5, 2.5, …

Fibonacci Sequences

Fibonacci sequences are named after Leonardo of Pisa (c 1200) who was the son of a certain Bonaccio hence Fi Bonaccio.

These sequences have the characteristic that each term is based upon some combination of two or more previous terms.

The original Fibonacci sequence goes 1, 1, 2, 3, 5, 8, ... Each term is the sum of the two previous terms.

One interesting fact about the Fibonacci terms is that if we work out the ratio between successive terms we find that the ratio converges to a very important number called the Golden Ratio or Divine Proportion, given the symbol Φ . It’s value is 1.618… This number is irrational.

There are some important results to do with Φ. For instance:

 We can also show that:

 =   

This strange thing on the right is called a continued fraction. Maths is full of interesting links and relationships…if you search for them.

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